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Chapter 2: Problem 129

For the following exercises, sketch the graph of each equation. $$q(x)=3$$

Short Answer

Expert verified
The graph is a horizontal line at \( y = 3 \).

Step by step solution

01

Analyze the equation

The given equation is \( q(x) = 3 \). This is a constant function where \( q(x) \) is equal to 3 for any value of \( x \). It indicates that the output does not change with different inputs.
02

Understand the graph of constant functions

For equations in the form of \( q(x) = c \), where \( c \) is a constant, the graph is a horizontal line across the y-axis at \( y = c \). This means that no matter the value of \( x \), the value of \( q(x) \) will always be 3.
03

Identify key points

Select a few points to plot the graph, such as \( (0, 3), (1, 3), (-1, 3) \). These points lie on the horizontal line \( y = 3 \).
04

Plot the graph

Draw a straight line horizontally across the y-axis at \( y = 3 \). Ensure that the line extends indefinitely in both the positive and negative directions along the x-axis, because \( q(x) = 3 \) holds for all \( x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Line
When you encounter a constant function like the one described by the equation \( q(x) = 3 \), one of the most noticeable characteristics is that its graph forms a horizontal line. A horizontal line means that all points on the graph have the same y-coordinate. For the function \( q(x) = 3 \), this horizontal line lies at \( y = 3 \) on the y-axis.
  • Every input \( x \) produces the same output \( 3 \), creating a line parallel to the x-axis.
  • The line does not slope upwards or downwards, which is the hallmark of constancy in a function.
These properties make constant functions easy to identify since regardless of how far you move along the x-axis, the line remains at the same height, reinforcing the idea of fixed output.
Graph Sketching
Sketching the graph of a constant function like \( q(x) = 3 \) involves noting a few straightforward steps that ensure accurate representation. First, rather than focusing on multiple changes in the x-values, shift attention to where the line levels out. With only one y-value to remember, sketching becomes much simpler:
  • Identify the horizontal line to place along the y-axis.
  • Choose key x-values, such as \( x = -1, 0, 1 \), to confirm they all produce the same y-value, which is \( 3 \).
  • Draw a straight line connecting these points along \( y = 3 \).
  • Extend this line indefinitely in both horizontal directions since \( q(x) \) remains 3 for all x.
This method not only highlights the graph's consistent nature but also enables quick validation that the visualization accurately describes the function.
Constant Value
In the context of the constant function \( q(x) = 3 \), the term 'constant value' refers to the unchanging output produced regardless of the input. This singular value, which is \( 3 \) here, illustrates:
  • Consistency across the entire domain of the function.
  • An unbroken output, meaning no fluctuations or variations occur no matter how x changes.
When you analyze a constant value, you're essentially looking at the defining feature of a constant function. It tells you that if you were to even attempt to plug in any number for x, from negative infinity to positive infinity, the result will always be 3. This is the unwavering characteristic that sets constant functions apart from other types of functions, which can vary depending on x.
Function Analysis
Function analysis of \( q(x) = 3 \) allows for understanding the core nature and implications of this mathematical expression. By breaking it down:
  • **Domain**: The domain of this function is all real numbers \( (-\infty, \infty) \), since \( x \) can take any value.
  • **Range**: The range is simply \( \{3\} \), as only a single output value is achievable regardless of the input.
  • **Slope**: This function has a slope of zero, reflecting the fact that there is no change in the y-value as x progresses.
  • **Intercepts**: The graph crosses the y-axis at \( y = 3 \). There are no x-intercepts because the line never reaches the x-axis.
This analysis highlights the unique simplicity of constant functions: predictable, invariant, and incredibly straightforward to visualize and interpret.

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